Properties

Label 20-9800e10-1.1-c1e10-0-0
Degree $20$
Conductor $8.171\times 10^{39}$
Sign $1$
Analytic cond. $8.61056\times 10^{18}$
Root an. cond. $8.84609$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·9-s − 12·11-s + 16·23-s + 12·29-s + 36·37-s + 24·43-s + 8·53-s + 40·67-s − 8·71-s − 4·79-s + 34·81-s + 96·99-s + 72·107-s + 44·109-s − 16·113-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 38·169-s + 173-s + ⋯
L(s)  = 1  − 8/3·9-s − 3.61·11-s + 3.33·23-s + 2.22·29-s + 5.91·37-s + 3.65·43-s + 1.09·53-s + 4.88·67-s − 0.949·71-s − 0.450·79-s + 34/9·81-s + 9.64·99-s + 6.96·107-s + 4.21·109-s − 1.50·113-s + 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.92·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{20} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 5^{20} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(2^{30} \cdot 5^{20} \cdot 7^{20}\)
Sign: $1$
Analytic conductor: \(8.61056\times 10^{18}\)
Root analytic conductor: \(8.84609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 2^{30} \cdot 5^{20} \cdot 7^{20} ,\ ( \ : [1/2]^{10} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(88.02048497\)
\(L(\frac12)\) \(\approx\) \(88.02048497\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + 8 T^{2} + 10 p T^{4} + 22 p T^{6} + 59 p T^{8} + 556 T^{10} + 59 p^{3} T^{12} + 22 p^{5} T^{14} + 10 p^{7} T^{16} + 8 p^{8} T^{18} + p^{10} T^{20} \)
11 \( ( 1 + 6 T + 40 T^{2} + 188 T^{3} + 827 T^{4} + 2652 T^{5} + 827 p T^{6} + 188 p^{2} T^{7} + 40 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
13 \( 1 + 38 T^{2} + 70 p T^{4} + 18256 T^{6} + 297797 T^{8} + 4047636 T^{10} + 297797 p^{2} T^{12} + 18256 p^{4} T^{14} + 70 p^{7} T^{16} + 38 p^{8} T^{18} + p^{10} T^{20} \)
17 \( 1 + 94 T^{2} + 278 p T^{4} + 159624 T^{6} + 3994333 T^{8} + 76875092 T^{10} + 3994333 p^{2} T^{12} + 159624 p^{4} T^{14} + 278 p^{7} T^{16} + 94 p^{8} T^{18} + p^{10} T^{20} \)
19 \( 1 + 78 T^{2} + 2997 T^{4} + 82408 T^{6} + 1981362 T^{8} + 41323348 T^{10} + 1981362 p^{2} T^{12} + 82408 p^{4} T^{14} + 2997 p^{6} T^{16} + 78 p^{8} T^{18} + p^{10} T^{20} \)
23 \( ( 1 - 8 T + 45 T^{2} - 208 T^{3} + 1516 T^{4} - 8336 T^{5} + 1516 p T^{6} - 208 p^{2} T^{7} + 45 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
29 \( ( 1 - 6 T + 74 T^{2} - 12 p T^{3} + 3305 T^{4} - 12940 T^{5} + 3305 p T^{6} - 12 p^{3} T^{7} + 74 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
31 \( 1 + 146 T^{2} + 10769 T^{4} + 16384 p T^{6} + 18108142 T^{8} + 569319516 T^{10} + 18108142 p^{2} T^{12} + 16384 p^{5} T^{14} + 10769 p^{6} T^{16} + 146 p^{8} T^{18} + p^{10} T^{20} \)
37 \( ( 1 - 18 T + 255 T^{2} - 2532 T^{3} + 20492 T^{4} - 137108 T^{5} + 20492 p T^{6} - 2532 p^{2} T^{7} + 255 p^{3} T^{8} - 18 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
41 \( 1 + 240 T^{2} + 28757 T^{4} + 2289600 T^{6} + 135644106 T^{8} + 6261316000 T^{10} + 135644106 p^{2} T^{12} + 2289600 p^{4} T^{14} + 28757 p^{6} T^{16} + 240 p^{8} T^{18} + p^{10} T^{20} \)
43 \( ( 1 - 12 T + 119 T^{2} - 624 T^{3} + 3546 T^{4} - 11848 T^{5} + 3546 p T^{6} - 624 p^{2} T^{7} + 119 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
47 \( 1 + 320 T^{2} + 48790 T^{4} + 4763690 T^{6} + 335304825 T^{8} + 17939282572 T^{10} + 335304825 p^{2} T^{12} + 4763690 p^{4} T^{14} + 48790 p^{6} T^{16} + 320 p^{8} T^{18} + p^{10} T^{20} \)
53 \( ( 1 - 4 T + 147 T^{2} - 760 T^{3} + 12592 T^{4} - 54184 T^{5} + 12592 p T^{6} - 760 p^{2} T^{7} + 147 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
59 \( 1 + 234 T^{2} + 35977 T^{4} + 3810176 T^{6} + 317161966 T^{8} + 20750042604 T^{10} + 317161966 p^{2} T^{12} + 3810176 p^{4} T^{14} + 35977 p^{6} T^{16} + 234 p^{8} T^{18} + p^{10} T^{20} \)
61 \( 1 + 380 T^{2} + 73905 T^{4} + 9478480 T^{6} + 879723030 T^{8} + 61495776552 T^{10} + 879723030 p^{2} T^{12} + 9478480 p^{4} T^{14} + 73905 p^{6} T^{16} + 380 p^{8} T^{18} + p^{10} T^{20} \)
67 \( ( 1 - 20 T + 377 T^{2} - 4168 T^{3} + 45876 T^{4} - 367304 T^{5} + 45876 p T^{6} - 4168 p^{2} T^{7} + 377 p^{3} T^{8} - 20 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
71 \( ( 1 + 4 T + 247 T^{2} + 336 T^{3} + 25806 T^{4} + 6744 T^{5} + 25806 p T^{6} + 336 p^{2} T^{7} + 247 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
73 \( 1 + 580 T^{2} + 160217 T^{4} + 27614864 T^{6} + 3281640790 T^{8} + 280650403224 T^{10} + 3281640790 p^{2} T^{12} + 27614864 p^{4} T^{14} + 160217 p^{6} T^{16} + 580 p^{8} T^{18} + p^{10} T^{20} \)
79 \( ( 1 + 2 T + 172 T^{2} - 368 T^{3} + 169 p T^{4} - 85252 T^{5} + 169 p^{2} T^{6} - 368 p^{2} T^{7} + 172 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} )^{2} \)
83 \( 1 + 414 T^{2} + 66037 T^{4} + 3463976 T^{6} - 330972942 T^{8} - 58619810764 T^{10} - 330972942 p^{2} T^{12} + 3463976 p^{4} T^{14} + 66037 p^{6} T^{16} + 414 p^{8} T^{18} + p^{10} T^{20} \)
89 \( 1 + 676 T^{2} + 215993 T^{4} + 43428112 T^{6} + 6118980374 T^{8} + 632067652568 T^{10} + 6118980374 p^{2} T^{12} + 43428112 p^{4} T^{14} + 215993 p^{6} T^{16} + 676 p^{8} T^{18} + p^{10} T^{20} \)
97 \( 1 + 542 T^{2} + 147830 T^{4} + 27425384 T^{6} + 3844015837 T^{8} + 420656703124 T^{10} + 3844015837 p^{2} T^{12} + 27425384 p^{4} T^{14} + 147830 p^{6} T^{16} + 542 p^{8} T^{18} + p^{10} T^{20} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.59232381128793649155622828634, −2.58973189459503510444151046168, −2.58573783936860092595384462703, −2.47742320231238474769591354579, −2.38474885575357466074348736584, −2.09255455887365842458690687227, −2.07962222167241745579182379487, −2.07060716202622496707762870600, −2.06600392911975809468412694656, −1.80261654403250365911803326341, −1.64323752432099931646175443530, −1.60871755316933894609837787864, −1.52951187350326269658854022103, −1.45372113284241171951379799766, −1.10986881773092054800309879361, −0.930666547294293914122475193551, −0.893700427286601717182806285288, −0.851859624565434548303525906734, −0.847544834865845520376875412355, −0.68287093404765646762682045381, −0.62337184204826408771196974194, −0.50702719639235333474666971426, −0.46162530189131879822819731305, −0.35061132203019140837799056427, −0.34701912374583725421930801445, 0.34701912374583725421930801445, 0.35061132203019140837799056427, 0.46162530189131879822819731305, 0.50702719639235333474666971426, 0.62337184204826408771196974194, 0.68287093404765646762682045381, 0.847544834865845520376875412355, 0.851859624565434548303525906734, 0.893700427286601717182806285288, 0.930666547294293914122475193551, 1.10986881773092054800309879361, 1.45372113284241171951379799766, 1.52951187350326269658854022103, 1.60871755316933894609837787864, 1.64323752432099931646175443530, 1.80261654403250365911803326341, 2.06600392911975809468412694656, 2.07060716202622496707762870600, 2.07962222167241745579182379487, 2.09255455887365842458690687227, 2.38474885575357466074348736584, 2.47742320231238474769591354579, 2.58573783936860092595384462703, 2.58973189459503510444151046168, 2.59232381128793649155622828634

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.